Weight functions for the determination of stress intensity factor and T‐stress for semi‐elliptical cracks in finite thickness plate

This paper presents the application of weight function method for the calculation of stress intensity factors (K) and T‐stress for surface semi‐elliptical crack in finite thickness plates subjected to arbitrary two‐dimensional stress fields. New general mathematical forms of point load weight functions for K and T have been formulated by taking advantage of the knowledge of a few specific weight functions for two‐dimensional planar cracks available in the literature and certain properties of weight function in general. The existence of the generalised forms of the weight functions simplifies the determination of specific weight functions for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalised weight function expression. These unknown parameters can be determined from reference stress intensity factor and T‐stress solutions. This method is used to derive the weight functions for both K and T for semi‐elliptical surface cracks in finite thickness plates, covering a wide range of crack aspect ratio (a/c) and relative depth (a/t) at any point along the crack front. The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

Universal features of weight functions for cracks in mode I

An analysis of known analytical and numerical weight functions for cracks in mode I has revealed that they all have a similar singular term and that it is possible to approximate them with one universal expression with three unknown parameters. The unknown parameters can be determined directly from reference stress intensity factor expressions without using the crack opening displacement function. The universal weight function expression, with suitable reference stress intensity factors, was used to derive the weight functions for internal and external radial cracks in a thick cylinder. These weight functions were then further used to calculate the stress intensity factors for radial cracks in a cylinder subjected to several nonlinear stress fields and were compared to available numerical data.     

COMPARISON OF SHORT AND LONG FATIGUE CRACK GROWTH IN 7075-T6 ALUMINUM

The fatigue crack growth rates of physically-short cracks (0.5 ≤a≤ 1.0 mm), intermediate cracks (1 < a≤ 2 mm) and long cracks (7 < a < 25 mm) were compared using SEN type tensile specimens in 7075-T6 aluminum alloy with load ratios, R, of 0.05, ? 1 and 0.5 under constant amplitude testing at room temperature. It was found that the short cracks grew much faster than long cracks based on applied δK with da/dN≤ 10^?7 m/cycle. Even the intermediate cracks grew faster than the long cracks below 10^?7 m/cycle. The transition crack lengths where similitude with δK existed was between 1 and 2 mm. Mean stress effects were similar for R= 0.05 and ? t, but R= 0.5 caused increased crack growth rates. The above differences are partially attributed to crack closure effects. Based upon plastic zone sizes, LEFM was justifiable with all the experiments.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

STRESS INTENSITY FACTORS AND WEIGHT FUNCTIONS FOR SEMI-ELLIPTICAL SURFACE CRACKS IN FINITE-THICKNESS PLATES UNDER TWO-DIMENSIONAL STRESS DISTRIBUTION

Abstract— A Fourier series approach is proposed to calculate stress intensity factors using weight functions for semi-elliptical surface cracks in flat plates subjected to two-dimensional stress distributions. The weight functions were derived from reference stress intensity factors obtained by three-dimensional finite element analyses. The close form weight functions derived are suitable for the calculation of stress intensity factors for semi-elliptical surface cracks in flat plates under two-dimensional stress distributions with the crack aspect ratio in the range of 0.1 ≤ a/c ≤ 1 and relative depth in the range of 0 ≤ a/t ≤ 0.8. Solutions were verified using several two-dimensional non-linear stress distributions; the maximum difference being 6%.     

Effect of curvature at the crack tip on the stress intensity factor for curved cracks

In this paper, the numerical solution of the hypersingular integral equation using the body force method in curved crack problems is presented. In the body force method, the stress fields induced by two kinds of standard set of force doublets are used as fundamental solutions. Then, the problem is formulated as a system of integral equations with the singularity of the form r ^–2. In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density functions and power series. The calculation shows that the present method gives rapidly converging numerical results for curved cracks under various geometrical conditions. In addition, a method of evaluation of the stress intensity factors for arbitrary shaped curved cracks is proposed using the approximate replacement to a simple straight crack.     

WEIGHT FUNCTIONS AND STRESS INTENSITY FACTORS FOR SEMI-ELLIPTICAL CRACKS IN T-PLATE WELDED JOINTS

Weight functions were derived for the deepest point and surface point of a semi-elliptical surface crack in T-plate joints with weld angles between 0 and 45°. These weight functions were derived from reference stress intensity factor solutions obtained from three-dimensional finite element calculations, and verified using stress intensity factors for different non-linear stress fields and for far-field tension and bending cases. The differences between the weight function predictions and the finite element data were less than 10%. They are suitable for semi-elliptical surface cracks with aspect ratios in the range 0.05 ≤ a/c ≤ 1, together with relative depths 0 ≤ a/t ≤ 0.6 and weld angles 0 ≤ φ ≤ 45°.     

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths (θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Weight function calculation for surface cracks using the line-spring model

A numerical method for calculating weight functions for surface cracks in plates and shells is proposed. Thick-shell finite elements are used to create the discrete model of a body with a through-wall flaw. Line-spring elements transform the through-wall flaw into a surface crack. A quadratic line-spring element is presented. Weight functions for some semielliptical surface cracks in a plate have been calculated. The weight functions obtained may be used for computing stress intensity factors related to two-dimensional stress fields at the crack surface.     

Stress-intensity factors for circular hollow section V-joints with a rack-plate chord

This study investigates the fatigue crack‐driving force, measured by the linear‐elastic stress‐intensity factors (SIFs), for a surface crack at the root of the welds in a thick‐walled, circular hollow section (CHS) V‐shape joint, typically installed in modern offshore jack‐up platforms. The primary (chord) member of the V‐joint consists of two half CHSs welded to both sides of a thick rack plate, while the secondary (brace) member adopts thick‐walled CHSs. The surface‐breaking crack considered in this study locates at the interface between the rack plate and the weld metal joining the half CHS, and represents an initial flaw introduced by lack of penetration in the welding procedure. The finite‐element model incorporates a very detailed, local crack‐front mesh in a global continuous mesh through a mesh‐tying procedure, which ensures displacement continuity between the independent master surface and the dependent slave surface. A simple plate model verifies the mesh‐tying procedure in computing the linear‐elastic stress‐intensity factors for two remote loading conditions. The computation of the stress‐intensity factors employs a linear‐elastic interaction integral approach. The comparison of the computed SIF values with a previous experimental measurement for a CHS T‐joint verifies the accuracy and feasibility of the interaction integral approach in computing SIF values for surface cracks in welded tubular connections. Subsequent numerical analysis on the gapped V‐joints examines the mixed‐mode SIF values for different loading conditions and includes an array of practical joint geometric parameters and crack sizes. The nondimensional mode I stress‐intensity factors generally increases with the following variations in the joint geometric parameters: an increase in the chord radius to the wall thickness ratio (γ=d₀/2t₀) , an increase in the brace diameter to the chord diameter ratio (β=d₁/d₀) , a decrease in the crack depth ratio (a/t) or an increase in the crack length c. The current study identifies a practical group of V‐joints that requires detailed treatment in the fatigue assessment procedure. These V‐joints adopt a large β ratio and demonstrate high mode‐mixity angles [ψ= tan^?1(K_II/K_I)] with correspondingly high mode I and mode II stress‐intensity factors.     

Partition of unity enrichment for bimaterial interface cracks

Partition of unity enrichment techniques are developed for bimaterial interface cracks. A discontinuous function and the two‐dimensional near‐tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity. This enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces. The crack‐tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation. The mixed‐mode (complex) stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized. Copyright © 2004 John Wiley & Sons, Ltd.     

Estimation of high‐temperature fracture parameters for small punch specimen with a surface crack

An outline of a newly proposed methodology for evaluating creep crack growth (CCG) parameters using cracked small‐punch (SP) specimens is explained. Three‐dimensional finite element analyses were performed to calculate the stress intensity factor along the crack front for a surface crack formed at the centre of a SP specimen. Effects of crack ratio, (a/t); crack aspect ratio, (a/c); and thickness of the specimen, (t), on the fracture parameters were studied. It was observed that the minimum variation of K‐value along the crack front can be achieved when a/c was 0.50 except the location very near the intersection of the crack and free surface. This condition is similar to the case of constant K‐values along the crack front of the conventional compact tension specimen. Thus, it can be argued that the SP specimen with a surface crack is a suitable specimen geometry for CCG testing. The proposed CCG test method was found to be practically applicable for the crack geometry of 0.10 to 0.30 of a/t with constant aspect ratio of 0.50. An estimation of the K and C_t‐parameter under the small scale creep condition was derived. Future work for further development of the suggested CCG testing is discussed.     

Crack growth behavior and fatigue-life prediction based on worst-case near-threshold data of a large crack for 9310 steel

ABSTRACT Both experimental and analytical investigations were conducted to study crack initiation and growth of small cracks, near‐threshold growth behavior of large cracks at constant R‐ratio/decreasing ΔK and constant K_max/decreasing ΔK, respectively, for 9310 steel. The results showed that a pronounced small‐crack effect was not observed even at R = ?1, small cracks initiated by a slip mechanism at strong slip sites. Worst‐case near‐threshold testing results for large cracks under several K_max values showed that an effect of K_max on the near‐threshold behavior does not exist in the present investigation. A worst‐case near‐threshold test for a large crack, i.e. constant K_max/decreasing ΔK test, can give a conservative prediction of growth behavior of naturally initiated small cracks. Using the worst‐case near‐threshold data for a large crack and crack‐tip constraint factor equations defined in the paper, Newman's total fatigue‐life prediction method was improved. The fatigue lives predicted by the improved method were in reasonable agreement with the experiments. A three‐dimensional (3D) weight function method was used to calculate stress‐intensity factors for a surface crack at a notch of the present SENT specimen (with r/w = 1/8) by using a finite‐element reference solution. The results were verified by limited finite‐element solutions, and agreed well with those calculated by Newman's stress‐intensity factor equations when the stress concentration factor of the present specimen was used in the equations.     

B.E. Analysis of a cracked cylindrical rod twisted by rotation of a flat annular stamp

Conclusions The use of suitable Green functions, in the BEM for axisymmetric bodies with cracks, yields accurate evaluation of the stress solution near the crack boundary. The singular character of the contact shearing stress between the stamp and the elastic medium influences the Mode III stress intensity factor for internal cracks more than for the external cracks. The validity of the numerical method has been verified through some characteristic examples. The accuracy of the results for K _IIIis displayed for internal and external cracks. Their interaction effects with the singularity due to the stamp are reflected as the geometric parameters are varied.Dedicated to G. Rieder     

A new weight function for one‐dimensional subsurface cracks under general loading

In the present study, weight functions (WFs) of a subsurface crack were derived by proposing a new general form for approximate one‐dimensional WF. The WFs coefficients were considered as a function of crack length to depth ratio and were obtained using reference stress intensity factors (SIFs) of 16 cracks under uniform, linear, and parabolic normal and shearing loadings. The verification was performed by comparison of the straight and coupled SIFs calculated by WF and finite element modelling under some complicated loadings. In conclusion, the WFs can be simply and effectively employed for evaluating the cracks under any complex stress distributions.     

A graph‐based postoptimization approach for covering arrays

Covering arrays (CAs) are combinatorial objects with interesting features that have practical applications such as experimental design and fault detection in hardware and software. We introduce a graph‐based postoptimization (GBPO) approach to reduce the size of CAs exploiting the redundancy in CAs previously constructed. To evidence the advantages of using GBPO, we have instantiated it with 2 sets of CAs: (1) 560 CAs of strength 2≤t≤6, alphabet 2≤v≤6, and parameters 3≤k≤32 generated by an optimized version of In‐Parameter‐Order‐Generalized (IPOG‐F) and GBPO improved all CAs, and 37 cases matched the best‐known upper bounds; and (2) 32 CAs of strength t=2, alphabet 3≤v≤6, and number of parameters 8≤k≤146; in this set, 16 cases were improved, and 16 cases were matched.     

Numerical analysis of 2-D crack propagation problems using the numerical manifold method

The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory.     

Singular boundary elements for the analysis of cracks in plane strain

Straight and curved cracks are modelled by direct formulation boundary elements, of geometry defined by Hermitian cubic shape functions. Displacement and traction are interpolated by the Hermitian functions, supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance r from the crack tip, and subdominant modes in which it is proportional to r^1·5. The singular functions extend over many boundary elements on each crack face. A nodal collocation scheme is used, in which additional boundary integral equations are obtained by differentiation of the equation obtained from Betti's theorem. The hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack. Examples are shown of the analysis of buried and edge cracks, to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements.     

Weight functions and stress intensity factors for the single crack round-ended straight lug

Investigations were performed for the round-ended straight attachment lug with a single crack emanating from the hole with the weight function method. The weight functions, covering the geometries from W/D=1.5 to W/D=4.0, were generated from the results obtained with a boundary element method using the approximate weight function technique. The results have been given both in the form of analytical weight functions and tabulated dimensionless stress intensity factors for simple normalized powers of the crack line loading. This is a simple straight forward procedure to calculate stress intensity factors once the crack line loading is approximated by a polynomial. The present method is also valid for deriving stress intensity factors and weight functions for general crack configurations.